# Fundamental Concepts Of Algebra

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FUNDAMENTAL CONCEPTS OF ALGEBRA

Donald L. White Department of Mathematical Sciences

Kent State University Release 3.0

January 12, 2009

Copyright c 2009 by D. L. White

Contents

1 Number Systems

1

1.1 The Basic Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Algebraic Properties of Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Formal Constructions of Number Systems . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Basic Number Theory

35

2.1 Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Divisibility of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Division Algorithm and Greatest Common Divisor . . . . . . . . . . . . . . . . . . . 44

2.4 Properties of the Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Prime Factorizations and Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.7 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.8 Congruence and Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Polynomials

84

3.1 Algebraic Properties of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2 Binomial Coeﬃcients and Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Divisibility and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.4 Synthetic Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5 Factors and Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.6 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.7 Irreducible Polynomials as Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A Trigonometry Review

139

B Answers to Selected Problems

141

i

Chapter 1

Number Systems

In this chapter we study the basic arithmetic and algebraic properties of the familiar number systems the integers, rational numbers, real numbers, and the possibly less familiar complex numbers. We will consider which algebraic properties these number systems have in common as well as the ways in which they diﬀer.

We will use the following notation to denote sets of numbers.

N = {1, 2, 3, . . .} = Natural Numbers

Z = {0, ±1, ±2, ±3, . . .} = Integers

Q=

a a, b ∈ Z, b = 0 = Rational Numbers b

R = Real Numbers

C = Complex Numbers

1.1 The Basic Number Systems

The ﬁrst numbers anyone learns about are the “counting numbers” or natural numbers 1, 2, 3, . . ., which we will denote by N. We eventually learn about the basic operations of addition and multiplication of natural numbers. These operations are examples of binary operations, that is, operations that combine any two natural numbers to obtain another natural number. Addition and multiplication of natural numbers satisfy some very nice properties, such as commutativity, associativity, and the distributive law, which we will study more formally in a later section.

The other familiar arithmetic operations of subtraction and division are really just the “inverse operations” of addition and multiplication, and will not be considered as basic operations. (Although multiplication of natural numbers is really just repeated addition, this is a much less obvious interpretation in other number systems.) If we only wish to consider the natural numbers, we quickly encounter problems with subtraction and division. These operations can be performed on pairs of natural numbers only in some cases. For example, 3 − 5 and 3 ÷ 5 are not natural numbers.

In order to be able to subtract, we introduce the number 0 and the “negatives” of the natural numbers to obtain the set of integers Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}. The number 0 acts as a

1

2

CHAPTER 1. NUMBER SYSTEMS

neutral element or identity element for addition, because for any integer a, a + 0 = a. The negative of any integer a acts as an inverse for a relative to addition, because a + (−a) = 0.

Of course, thinking of 0 and −a in terms of addition leads to some interesting questions. Why is 0 times any number equal to 0? Why is the product of two negative numbers a positive number? More generally, what does multiplication by a negative number really mean? Is it still repeated addition? We will return to these questions later.

The operations of addition and multiplication in Z still satisfy the same properties as in the set of natural numbers N, but Z has an identity element and inverses for addition. This allows us to subtract any integer from any other and obtain another integer.

Division can still only be performed on certain pairs of integers, however. Although the number 1 acts as a neutral element or identity element for multiplication in Z, since 1 · a = a for every integer a, the set Z does not have inverses relative to multiplication for most of its elements. In order to be able to divide, we must introduce fractions and obtain the set Q of rational numbers, with operations deﬁned as follows.

Deﬁnition 1.1.1 The set Q of rational numbers is the set of all quotients of integers (i.e.,

fractions),

Q = a a, b ∈ Z, b = 0 , b

and we deﬁne

i. a = a if and only if ab = ba , bb

ii. a + c = ad + bc ,

bd

bd

iii. a · c = ac . b d bd

Note

that

since

a

=

a 1

for

an

integer

a,

every

integer

is

also

a

rational

number

and

we

have

Z

⊆

Q.

The operations of addition and multiplication in Q still satisfy all of the properties as in the set of

integers Z.

If

q

=

0

is

a

rational

number,

say

q

=

ab ,

then

a

=

0,

and

r

=

b a

is

also

a

rational

number.

Since

q · r = a · b = ab = 1, b a ba

the rational number

b a

is an inverse for

a b

relative to multiplication.

That is, if q =

a b

= 0 is a

rational number, the multiplicative inverse of q is q−1 = ab , the reciprocal of q.

A proper construction of the set R of real numbers requires tools from analysis beyond the

scope of this text. A less rigorous description of R in terms of decimal expansions will suﬃce for

our purposes. We will ﬁrst recall the basics of decimal expansions and discuss decimal expansions

of rational numbers.

A positive integer m can always be written in its decimal form and expressed as a sum of

multiples of non-negative powers of 10:

m = nknk−1 . . . n2n1n0 = nk · 10k + nk−1 · 10k−1 + · · · + n2 · 102 + n1 · 101 + n0 · 100.

1.1. THE BASIC NUMBER SYSTEMS

3

Similarly, a positive number r < 1 with a terminating decimal expansion can be written as a sum of multiples of negative powers of 10:

r = 0.d1d2 . . . dk−1dk = d1 · 10−1 + d2 · 10−2 + · · · + dk−1 · 10−(k−1) + dk · 10−k.

If the decimal expansion does not terminate, then r is an “inﬁnite sum” of multiples of negative

powers of 10: r = 0.d1d2d3 . . . = d1 · 10−1 + d2 · 10−2 + d3 · 10−3 + · · · .

In general, a (real) number can be written as a ﬁnite sum of multiples of non-negative powers of 10

plus a (possibly) inﬁnite sum of multiples of negative powers of 10, and the coeﬃcients in this sum

are the digits in the decimal expansion of the number.

You may recall that the set of rational numbers deﬁned in Deﬁnition 1.1.1 can also be described

in terms of decimal expansions. The rational numbers are precisely those (real) numbers with ter-

minating or repeating decimal expansions. For example, 3/8 = 0.375 or 9/7 = 1.285714285714 . . . =

1.285714.

In order to verify this characterization of rational numbers, we must show that if

a b

is

any

rational

number,

then

the

decimal

expansion

of

a b

either

terminates

or

is

a

repeating

decimal,

and that every terminating or repeating decimal is the decimal expansion of a rational number ab .

We

will

ﬁrst

verify

that

every

rational

number

a b

has

either

a

terminating

or

repeating

decimal

expansion.

First

note

that

we

may

assume

a b

is

positive

and

a

<

b.

(Why?)

The

decimal

expansion

of

a b

is

obtained

by

performing

the

long

division

a ÷ b.

In the algorithm for long division, we place a decimal point to the right of a and append just

enough zeros to obtain a number larger than a (without the decimal). We then divide, placing the

quotient above the division sign and obtaining a remainder r with 0 r < b. (This is possible by

the Division Algorithm, which we will study formally in §2.3.) The algorithm is then repeated to

divide r by b and so on, as demonstrated for 3 ÷ 17 in the following example:

0. 1 7 6 4 . . . 17 ) 3. 0

17 1 30 1 19

110 102

80 ...

The boldface numbers in the example are the remainders. If a remainder of 0 is obtained at some stage, then the decimal expansion terminates, as for 38 :

0. 3 7 5 8 ) 3. 0

24 60 56 40 40 0

If the remainder is never zero, then each remainder r satisﬁes 1 r b − 1. Thus there are only ﬁnitely many diﬀerent possible remainders, and at some point a remainder must repeat. Once a

4

CHAPTER 1. NUMBER SYSTEMS

remainder repeats, the same sequence of quotients and remainders must repeat forever, and the decimal expansion is repeating, as for 574 :

0. 1 2 9 6 2 9 6 . . . 54 ) 7. 0

54 1 60 1 08

520 486

340 324

160 108

520 486

340 324

16 ...

Notice that 16 is the ﬁrst remainder to repeat in this example and the corresponding quotient, 2, is the start of the repeating decimal.

Exploration: Under what conditions on a fraction a/b in lowest terms will the decimal expansion be terminating? Investigate this question by searching in number theory texts or Internet sources.

Conversely, we must verify that every terminating or repeating decimal expansion is the decimal

expansion of a rational number. Again, we may assume the decimal number is positive and less

than one. (Why?) A terminating decimal with k decimal places, say .d1d2 . . . dk, can be written as

the fraction

d1d2 . . . dk . 10k

We say that the period of a repeating decimal is k if the length of the shortest repeating

sequence of digits is k. For example, the period of 0.454545 . . . = 0.45 is 2 and the period of

0.1234563456 . . . = 0.123456 is 4.

A repeating decimal of period k, say

R = 0.d1d2 . . . djr1r2 . . . rk,

can be expressed as a fraction, that is, a rational number, as follows. First, multiply R by 10k to obtain 10kR. This has the eﬀect of moving the decimal point k places to the right, or equivalently, shifting the digits of R k places to the left. Because the period of R is k, the digits of R and 10kR will be the same after some decimal place. Thus the number

10kR − R = (10k − 1)R

will be a terminating decimal, hence equal to some fraction T . Therefore (10k − 1)R = T and R= T 10k − 1

is a rational number.

1.1. THE BASIC NUMBER SYSTEMS

5

Example: Write the repeating decimal R = 0.12345345 . . . = 0.12345 as a fraction. The period of R is 3, so we calculate 103R = 1000R:

R = 0 . 12 345 345 1000R = 123 . 45 345 345,

thus and so

1000R − R = 999R = 123.33 = 12333 100

R = 12333 = 12333 = 4111 . 999 · 100 99900 33300

The set R of real numbers consists of all possible decimal expansions. We have shown that the rational numbers are precisely those real numbers with either terminating or repeating decimal expansions. As there are clearly decimal expansions that are not repeating (for example, 0.01011011101111 . . .), not all real numbers are rational. Those real numbers that do not have terminating or repeating decimal expansions, and therefore are not rational, are called irrational numbers. Thus R consists of the rational numbers along with the irrational numbers.

The irrational numbers are real numbers that cann√ot be expressed as a quotient of two integers. Some well-known examples of irrational numbers are 2, e, and π, but there are many others. In fact, in a sense that can be made precise, most real numbers are irrational.

For computational purposes, we usually approximate irrational numbers by rational numbers. For example, your calculator probably uses the approximation 3.141592654 (a rational number) for π in calculations.

This approximation is suﬃciently accurate for most purposes, but π, or any other irrational number, can be approximated to within any desired degree of accuracy by a rational number. Probably the easiest way to see this is to note that truncating the decimal expansion of the irrational number results in a terminating decimal, hence a rational number, and the greater the number of decimal places used, the closer the approximation will be.

In particular, if I is an irrational number and R is the rational number obtained by using the digits of I to the left of the decimal and the ﬁrst k digits of I to the right of the decimal, then 0 < I −R < 10−k. Thus R approximates I to within 10−k. For example, if R = 3.141592653589793, then

0 < π − R < 10−15 = 0.000000000000001.

Finally, we note that it is not possible to determine whether a number is rational or irrational from any terminating decimal approximation. For example, a calculator with a 10-digit display will show e ≈ 2.718281828, which certainly appears to be a repeating decimal, although e is in fact irrational. (The next digit in the decimal expansion of e is 4.) On the other hand, the calculator shows 1/17 ≈ .0588235294, which shows no evidence of repetition, although 1/17 is clearly rational. To verify that a given number I is irrational, it is necessary to prove that there cannot be integers a and b such that I = a/b.

6

CHAPTER 1. NUMBER SYSTEMS

§1.1 Exercises

1. Use long division to ﬁnd the repeating decimal expansion of the following rational numbers. Show your work on the long division.

(a) 5 101

(c) 17 135

(b) 47 110

(d) 5 14

2. Convert the following repeating decimal expansions to fractions in lowest terms.

(a) 0.393939 . . . = 0.39 (b) 4.302302302 . . . = 4.302 (c) 57.13478478478 . . . = 57.13478 (d) 102.102537253725372 . . . = 102.1025372

3. Show that 1 = 0.999999 . . . = 0.9.

4. Explain why the period of a rational number a/b with a repeating decimal is at most b − 1.

5.

By

Deﬁnition

1.1.1,

two

fractions

a b

and

a b

are equal if and only if ab

= ba .

Show that if

a b

=

a b

and

c d

=

c d

,

then

(a) a + c = a + c , that is, ad + bc = a d + b c and

bd b d

bd

bd

(b) a · c = a · c , that is, ac = a c .

bd b d

bd b d

(This exercise shows that addition and multiplication of rational numbers are “well-deﬁned,” so that the sum or product of two rational numbers does not depend on the particular representation of the numbers as fractions.)

6. Write a paragraph with your explanation to a middle school or high school student as to why 0 times any number is 0.

7. Write a paragraph with your explanation to a middle school or high school student as to why the product of two negative numbers is a positive number.

Donald L. White Department of Mathematical Sciences

Kent State University Release 3.0

January 12, 2009

Copyright c 2009 by D. L. White

Contents

1 Number Systems

1

1.1 The Basic Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Algebraic Properties of Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Formal Constructions of Number Systems . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Basic Number Theory

35

2.1 Principle of Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 Divisibility of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Division Algorithm and Greatest Common Divisor . . . . . . . . . . . . . . . . . . . 44

2.4 Properties of the Greatest Common Divisor . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 Prime Factorizations and Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.7 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.8 Congruence and Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Polynomials

84

3.1 Algebraic Properties of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2 Binomial Coeﬃcients and Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Divisibility and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.4 Synthetic Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5 Factors and Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.6 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.7 Irreducible Polynomials as Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A Trigonometry Review

139

B Answers to Selected Problems

141

i

Chapter 1

Number Systems

In this chapter we study the basic arithmetic and algebraic properties of the familiar number systems the integers, rational numbers, real numbers, and the possibly less familiar complex numbers. We will consider which algebraic properties these number systems have in common as well as the ways in which they diﬀer.

We will use the following notation to denote sets of numbers.

N = {1, 2, 3, . . .} = Natural Numbers

Z = {0, ±1, ±2, ±3, . . .} = Integers

Q=

a a, b ∈ Z, b = 0 = Rational Numbers b

R = Real Numbers

C = Complex Numbers

1.1 The Basic Number Systems

The ﬁrst numbers anyone learns about are the “counting numbers” or natural numbers 1, 2, 3, . . ., which we will denote by N. We eventually learn about the basic operations of addition and multiplication of natural numbers. These operations are examples of binary operations, that is, operations that combine any two natural numbers to obtain another natural number. Addition and multiplication of natural numbers satisfy some very nice properties, such as commutativity, associativity, and the distributive law, which we will study more formally in a later section.

The other familiar arithmetic operations of subtraction and division are really just the “inverse operations” of addition and multiplication, and will not be considered as basic operations. (Although multiplication of natural numbers is really just repeated addition, this is a much less obvious interpretation in other number systems.) If we only wish to consider the natural numbers, we quickly encounter problems with subtraction and division. These operations can be performed on pairs of natural numbers only in some cases. For example, 3 − 5 and 3 ÷ 5 are not natural numbers.

In order to be able to subtract, we introduce the number 0 and the “negatives” of the natural numbers to obtain the set of integers Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}. The number 0 acts as a

1

2

CHAPTER 1. NUMBER SYSTEMS

neutral element or identity element for addition, because for any integer a, a + 0 = a. The negative of any integer a acts as an inverse for a relative to addition, because a + (−a) = 0.

Of course, thinking of 0 and −a in terms of addition leads to some interesting questions. Why is 0 times any number equal to 0? Why is the product of two negative numbers a positive number? More generally, what does multiplication by a negative number really mean? Is it still repeated addition? We will return to these questions later.

The operations of addition and multiplication in Z still satisfy the same properties as in the set of natural numbers N, but Z has an identity element and inverses for addition. This allows us to subtract any integer from any other and obtain another integer.

Division can still only be performed on certain pairs of integers, however. Although the number 1 acts as a neutral element or identity element for multiplication in Z, since 1 · a = a for every integer a, the set Z does not have inverses relative to multiplication for most of its elements. In order to be able to divide, we must introduce fractions and obtain the set Q of rational numbers, with operations deﬁned as follows.

Deﬁnition 1.1.1 The set Q of rational numbers is the set of all quotients of integers (i.e.,

fractions),

Q = a a, b ∈ Z, b = 0 , b

and we deﬁne

i. a = a if and only if ab = ba , bb

ii. a + c = ad + bc ,

bd

bd

iii. a · c = ac . b d bd

Note

that

since

a

=

a 1

for

an

integer

a,

every

integer

is

also

a

rational

number

and

we

have

Z

⊆

Q.

The operations of addition and multiplication in Q still satisfy all of the properties as in the set of

integers Z.

If

q

=

0

is

a

rational

number,

say

q

=

ab ,

then

a

=

0,

and

r

=

b a

is

also

a

rational

number.

Since

q · r = a · b = ab = 1, b a ba

the rational number

b a

is an inverse for

a b

relative to multiplication.

That is, if q =

a b

= 0 is a

rational number, the multiplicative inverse of q is q−1 = ab , the reciprocal of q.

A proper construction of the set R of real numbers requires tools from analysis beyond the

scope of this text. A less rigorous description of R in terms of decimal expansions will suﬃce for

our purposes. We will ﬁrst recall the basics of decimal expansions and discuss decimal expansions

of rational numbers.

A positive integer m can always be written in its decimal form and expressed as a sum of

multiples of non-negative powers of 10:

m = nknk−1 . . . n2n1n0 = nk · 10k + nk−1 · 10k−1 + · · · + n2 · 102 + n1 · 101 + n0 · 100.

1.1. THE BASIC NUMBER SYSTEMS

3

Similarly, a positive number r < 1 with a terminating decimal expansion can be written as a sum of multiples of negative powers of 10:

r = 0.d1d2 . . . dk−1dk = d1 · 10−1 + d2 · 10−2 + · · · + dk−1 · 10−(k−1) + dk · 10−k.

If the decimal expansion does not terminate, then r is an “inﬁnite sum” of multiples of negative

powers of 10: r = 0.d1d2d3 . . . = d1 · 10−1 + d2 · 10−2 + d3 · 10−3 + · · · .

In general, a (real) number can be written as a ﬁnite sum of multiples of non-negative powers of 10

plus a (possibly) inﬁnite sum of multiples of negative powers of 10, and the coeﬃcients in this sum

are the digits in the decimal expansion of the number.

You may recall that the set of rational numbers deﬁned in Deﬁnition 1.1.1 can also be described

in terms of decimal expansions. The rational numbers are precisely those (real) numbers with ter-

minating or repeating decimal expansions. For example, 3/8 = 0.375 or 9/7 = 1.285714285714 . . . =

1.285714.

In order to verify this characterization of rational numbers, we must show that if

a b

is

any

rational

number,

then

the

decimal

expansion

of

a b

either

terminates

or

is

a

repeating

decimal,

and that every terminating or repeating decimal is the decimal expansion of a rational number ab .

We

will

ﬁrst

verify

that

every

rational

number

a b

has

either

a

terminating

or

repeating

decimal

expansion.

First

note

that

we

may

assume

a b

is

positive

and

a

<

b.

(Why?)

The

decimal

expansion

of

a b

is

obtained

by

performing

the

long

division

a ÷ b.

In the algorithm for long division, we place a decimal point to the right of a and append just

enough zeros to obtain a number larger than a (without the decimal). We then divide, placing the

quotient above the division sign and obtaining a remainder r with 0 r < b. (This is possible by

the Division Algorithm, which we will study formally in §2.3.) The algorithm is then repeated to

divide r by b and so on, as demonstrated for 3 ÷ 17 in the following example:

0. 1 7 6 4 . . . 17 ) 3. 0

17 1 30 1 19

110 102

80 ...

The boldface numbers in the example are the remainders. If a remainder of 0 is obtained at some stage, then the decimal expansion terminates, as for 38 :

0. 3 7 5 8 ) 3. 0

24 60 56 40 40 0

If the remainder is never zero, then each remainder r satisﬁes 1 r b − 1. Thus there are only ﬁnitely many diﬀerent possible remainders, and at some point a remainder must repeat. Once a

4

CHAPTER 1. NUMBER SYSTEMS

remainder repeats, the same sequence of quotients and remainders must repeat forever, and the decimal expansion is repeating, as for 574 :

0. 1 2 9 6 2 9 6 . . . 54 ) 7. 0

54 1 60 1 08

520 486

340 324

160 108

520 486

340 324

16 ...

Notice that 16 is the ﬁrst remainder to repeat in this example and the corresponding quotient, 2, is the start of the repeating decimal.

Exploration: Under what conditions on a fraction a/b in lowest terms will the decimal expansion be terminating? Investigate this question by searching in number theory texts or Internet sources.

Conversely, we must verify that every terminating or repeating decimal expansion is the decimal

expansion of a rational number. Again, we may assume the decimal number is positive and less

than one. (Why?) A terminating decimal with k decimal places, say .d1d2 . . . dk, can be written as

the fraction

d1d2 . . . dk . 10k

We say that the period of a repeating decimal is k if the length of the shortest repeating

sequence of digits is k. For example, the period of 0.454545 . . . = 0.45 is 2 and the period of

0.1234563456 . . . = 0.123456 is 4.

A repeating decimal of period k, say

R = 0.d1d2 . . . djr1r2 . . . rk,

can be expressed as a fraction, that is, a rational number, as follows. First, multiply R by 10k to obtain 10kR. This has the eﬀect of moving the decimal point k places to the right, or equivalently, shifting the digits of R k places to the left. Because the period of R is k, the digits of R and 10kR will be the same after some decimal place. Thus the number

10kR − R = (10k − 1)R

will be a terminating decimal, hence equal to some fraction T . Therefore (10k − 1)R = T and R= T 10k − 1

is a rational number.

1.1. THE BASIC NUMBER SYSTEMS

5

Example: Write the repeating decimal R = 0.12345345 . . . = 0.12345 as a fraction. The period of R is 3, so we calculate 103R = 1000R:

R = 0 . 12 345 345 1000R = 123 . 45 345 345,

thus and so

1000R − R = 999R = 123.33 = 12333 100

R = 12333 = 12333 = 4111 . 999 · 100 99900 33300

The set R of real numbers consists of all possible decimal expansions. We have shown that the rational numbers are precisely those real numbers with either terminating or repeating decimal expansions. As there are clearly decimal expansions that are not repeating (for example, 0.01011011101111 . . .), not all real numbers are rational. Those real numbers that do not have terminating or repeating decimal expansions, and therefore are not rational, are called irrational numbers. Thus R consists of the rational numbers along with the irrational numbers.

The irrational numbers are real numbers that cann√ot be expressed as a quotient of two integers. Some well-known examples of irrational numbers are 2, e, and π, but there are many others. In fact, in a sense that can be made precise, most real numbers are irrational.

For computational purposes, we usually approximate irrational numbers by rational numbers. For example, your calculator probably uses the approximation 3.141592654 (a rational number) for π in calculations.

This approximation is suﬃciently accurate for most purposes, but π, or any other irrational number, can be approximated to within any desired degree of accuracy by a rational number. Probably the easiest way to see this is to note that truncating the decimal expansion of the irrational number results in a terminating decimal, hence a rational number, and the greater the number of decimal places used, the closer the approximation will be.

In particular, if I is an irrational number and R is the rational number obtained by using the digits of I to the left of the decimal and the ﬁrst k digits of I to the right of the decimal, then 0 < I −R < 10−k. Thus R approximates I to within 10−k. For example, if R = 3.141592653589793, then

0 < π − R < 10−15 = 0.000000000000001.

Finally, we note that it is not possible to determine whether a number is rational or irrational from any terminating decimal approximation. For example, a calculator with a 10-digit display will show e ≈ 2.718281828, which certainly appears to be a repeating decimal, although e is in fact irrational. (The next digit in the decimal expansion of e is 4.) On the other hand, the calculator shows 1/17 ≈ .0588235294, which shows no evidence of repetition, although 1/17 is clearly rational. To verify that a given number I is irrational, it is necessary to prove that there cannot be integers a and b such that I = a/b.

6

CHAPTER 1. NUMBER SYSTEMS

§1.1 Exercises

1. Use long division to ﬁnd the repeating decimal expansion of the following rational numbers. Show your work on the long division.

(a) 5 101

(c) 17 135

(b) 47 110

(d) 5 14

2. Convert the following repeating decimal expansions to fractions in lowest terms.

(a) 0.393939 . . . = 0.39 (b) 4.302302302 . . . = 4.302 (c) 57.13478478478 . . . = 57.13478 (d) 102.102537253725372 . . . = 102.1025372

3. Show that 1 = 0.999999 . . . = 0.9.

4. Explain why the period of a rational number a/b with a repeating decimal is at most b − 1.

5.

By

Deﬁnition

1.1.1,

two

fractions

a b

and

a b

are equal if and only if ab

= ba .

Show that if

a b

=

a b

and

c d

=

c d

,

then

(a) a + c = a + c , that is, ad + bc = a d + b c and

bd b d

bd

bd

(b) a · c = a · c , that is, ac = a c .

bd b d

bd b d

(This exercise shows that addition and multiplication of rational numbers are “well-deﬁned,” so that the sum or product of two rational numbers does not depend on the particular representation of the numbers as fractions.)

6. Write a paragraph with your explanation to a middle school or high school student as to why 0 times any number is 0.

7. Write a paragraph with your explanation to a middle school or high school student as to why the product of two negative numbers is a positive number.

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